5.3 Generic plateau inflation
5.3.1 Analytic predictions
We now demonstrate how coupling a scalar field non-minimally to gravity may realize an approximately shift-symmetric Einstein frame potential employing a minimal set of assumptions. The aim of this section is to explicitly show the robustness of the inflationary potential from an arbitrary number of higher order terms.
To that end, consider the non-minimal coupling or frame function as well as the potential to be given by arbitrary series with the only requirement that the Jordan frame potential and the square of the frame function share the order of
5.3. GENERIC PLATEAU INFLATION 89 their first zero for φ≥0.35 In other words, we require the Jordan frame potential to have a minimum and the frame function to contain a term linear in the Jordan frame field φ. We thus write the setup in full generality as
Ω(φ) = 1 +ξX
n=1
anφn, VJ(φ) =λX
m=2
bmφm, (5.19) where we have kept the factor λ to be consistent with the original work but will assume it to take a natural value of . O(1). We have further omitted to specify the cut-off of either series as it will not play a role in the subsequent analysis. In principle, both series may contain an infinite number of terms. The assumption that the Jordan frame field φ is stabilised translates into the requirement b2 > 0.
We further, as already stated, take a1 >0, i.e. assume the non-minimal coupling to be approximated by a polynomial series expansion around the minimum of the potentialφ = 0. This also implies that the canonical Einstein frame inflatonϕand Jordan frame field φ decrease correspondingly, i.e. dϕ/dφ > 0. This is necessary for the canonical field ϕ not to have a runaway direction in the potential which might prevent the inflaton from gracefully exiting slow roll.
In the Jordan frame, higher order terms are sub-leading if the Jordan frame field φ remains sub-Planckian. From this, two questions arise. First, is it possible to generate a sufficient amount of e-folds within one Planck distance in the Jordan frame field. Secondly, how does this argument carry over to the Einstein frame and the non-minimal coupling strength ξ.
For set-up (5.19), and for now assuming to be in the regime φ <1, the expres-sion for the number of e-folds of (5.5) obtains corrections as
Ne ∼ 3
4Ω− b3Ω 8b2a1
Ω2 ξ
+O(2) Ω2
ξ
= Ω 3
4 − b3 8b2a1
Ω2 ξ
+. . .
, (5.20) which may be understood as an expansion in Ω2/ξ. Given that we seek a mech-anism yielding inflationary dynamics compatible with PLANCK, we require the corrections to the leading order term of the above to be sub-dominant. This is the case for Ω2 << ξ, and hence leads to a self-consistent expansion. Setting
35We are only considering zeroes and not poles in this set-up.
Ne ≡NCM B ∼ O(60), where NCM B denotes the number of e-folds at horizon exit of scales now observable through the CMB, we thus find that the lower bound on the non-minimal coupling strength for generating a sufficient amount of inflation within ∆φ <1 is
ξ >O(Ne2). (5.21)
The following discussion hence assumes this lower bound. This result is crucial as it demonstrates that the predictions of the general ansatz (5.19) begin to converge towards the universal predictions (5.9) when the non-minimal coupling strength is of the order of the amount of e-folds minimally required.
In the regime φ < 1 the first zeros in both series of (5.19) are leading. We hence infer that non-canonical field and frame function may be related as
φ ∼ 1
a1ξ(Ω−1). (5.22)
It is readily verified from ansatz (5.19) that the Einstein frame potential then becomes
VE = λ a21ξ2
1− 1
Ω 2"
b2+X
k=1
bk+2
Ω−1 a1ξ
k#
. (5.23)
First we note that as long as the summand is less than unity, the series of all corrections converges. We conclude that makingξsufficiently large can, regardless of aninfinite tower of higher order corrections with order one coefficients, induce a Starobinsky-like inflationary plateau over a finite field range. This is the Einstein frame manifestation of the fact that during inflation, φ < 1 and hence all higher order terms in the Jordan frame potential are sub-leading.
In other words, the effect of higher order terms can simply be pushed far away in field space by sufficiently enlarging the non-minimal coupling strengthξ. Thus the inflationary dynamics are independent of whether or not the tower of higher order corrections is truncated at some order. Hence we see that, given inflation occurs for the non-canonical field φ < 1 which can be ensured via having ξ & O(Ne2), the set-up is independent of the truncation of the potential, and the non-minimal coupling strength ξ therefore protects a finite plateau. Another way to look at this is the following: The expression for the displacement of the non-canonical
5.3. GENERIC PLATEAU INFLATION 91
ns
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
r
10-4 10-3 10-2 10-1 100 101
MV=4 MV=10
ns
0.9 0.92 0.94 0.96 0.98 1 1.02
MV=3 MV=4 MV=6 MV=8 MV=10 MV=12
Figure 5.3: Left: The plot shows a scatter plot with ξ = 104, MΩ = 1 and MV = 10 containing 5000 trajectories. The red dot represents the Starobinsky point ns(50) = 0.962, r = 0.004. The green line in the upper left displays chaotic signatures. The slope of the line in the vicinity of the Starobinsky point is, as predicted, δr/δns∼0.1. Right: An ns density plot, on a linear scale, for different values of MV. The plot peaks around the Starobinsky value.
field during inflation reads to leading order
∆φ∼ 1
ξ∆Ω, (5.24)
where ∆Ω denotes the change of the frame function between horizon exit of CMB scales and the end of inflation and is typically ∆Ω ∼ O(60). One immediately understands that an increase in ξ can force ∆φ sub-Planckian as ∆Ω is fixed through NCM B.
To obtain a value for ξ that ensures the corrections to be sufficiently far away from the minimum of the inflaton potential and to have inflation matching obser-vations by PLANCK, it is most useful to study the inflationary observables and their dependence on the infinite tower of higher order terms. To leading order, the
expressions for the inflationary observables ns and r of (5.19) and thus (5.23) are ns = 1− 2
Ne
+ 8κ6b3 b2
Ne a1ξ
+O(2) 1
Ne
, Ne a1ξ
,
r = 12
Ne2 + 32κ4b3 b2
1 a1ξ
+ 32κ8 b3
b2 2
Ne a1ξ
2
+O(3) 1
Ne, Ne
a1ξ
, (5.25)
where again κ=p
2/3. Expressions (5.25) are expansions in 1/Ne and Ne/(a1ξ).
For the spectral index ns, the leading order terms are the linear contributions of the 1/Neand theNe/(a1ξ) expansions. For the tensor to scalar ratior, the leading order terms are the quadratic and bilinear expressions of both expansions. Sub-leading terms stem from cross terms and higher orders in 1/(a1ξ) andNe/(a1ξ) and are denoted byO(n). Note that we have omitted the subleading corrections of [49], i.e. higher order terms in 1/Ne, for clarity. Forns and r to be dominated respec-tively by the linear and quadratic term in 1/Ne, i.e. for prolonging the Einstein frame potential’s intermediate plateau, we quickly identify that
ξ >O Ne2
, (5.26)
provideda1 ∼ O(1). This hence marks the onset of a convergence of the inflation-ary predictions towards the values measured.
Considering that we eventually seek to study models with random an, bm, ex-pressions (5.25) predict a range ofns, r pairs where however the slope of a scatter plot r vs. ns ought to be independent of the random draws as the leading order corrections, i.e. the next to leading order terms, both come with the samea1, b2, b3 dependence. Thus consider the ratio of the next to leading order terms
δr δns = 6
Ne ∼ O(0.1) (5.27)
forNe∼NCM B. Note that this implies a scatter plot of (ns, r) pairs to demonstrate a 1/Ne scaling in the slope at the Starobinsky point, i.e. scatter plots for different
5.3. GENERIC PLATEAU INFLATION 93 values ofNe will show a different slope. Generally, this predicts that in the vicinity of the Starobinsky point, there will be signatures to the bottom left and top right, roughly aligned with a slope of O(0.1) for Ne ∼ NCM B. In other words, given (5.25), we see that when b3 is positive the effect of steepening corrections is to increase the spectral index and the tensor to scalar ratio. Thus if coefficients are arbitrary and a largeξ not automatically protects the plateau fully against higher order terms, we expect signatures to appear in the ns, r plot that are higher than and to the right of the Starobinsky point. This corresponds to an upward curve in the potential plateau, roughly indicating the onset of a monomial dominated chaotic phase. Allowing for the first higher order coefficient b3 to be negative, potential (5.23) may obtain a hilltop feature. This means that the first order correction of (5.25) now comes with a minus sign. We hence expect signatures to appear below and to the left of the Starobinsky point. The left panels of Figures 5.3 and 5.4 nicely depict this behaviour. Note that a1 and b2 have to be positive to guaranty the positivity of the frame function and the potential around the minimum.
For higher order terms not to spoil the value of ns observed by PLANCK, i.e.
for the observables to enter the Planck contours, we consider the 2-σ bound by PLANCK of δns<0.008 at Ne = 55 and find, given a1, b2, b3 ∼ O(1),
ξ &O 104
. (5.28)
This hence sets, given order one coefficients, a lower bound on the non-minimal coupling strength ξ to realize observationally viable slow-roll inflation.
A few comments are in order. Not only does a fixed value of ξ prevent all higher order terms from becoming important before very roughly
ϕ∼κ−1log (a1ξ) (5.29)
but also is the value of ξ obtained from the requirement of matching the observed spectral index ns similar to the value needed to match COBE normalization (pro-vided λ . O(1)). Thus two independent observational indications - in technical terms the spectral index ns and the amplitude As - hint towards an otherwise ad
hoc value of the theory’s parameter. The length and the height of the inflationary plateau are correctly set by the single parameterξ. The rare case that coefficients an, bm are randomly drawn such that bi b2, b3 for i > 2 and hence that higher order terms evade the ξ induced flattening will be discussed in appendix B.